I've heard that if you have a ribbon wrapped around the equator of the
earth, and you want to increase its length so that it floats 1" above
the earth all the way around, you only need to add 6.28" to the
ribbon. Is that really true? How is it possible?

How can I find the equation for the radius of a 'racing circle' (the
fastest path a racecar can take through the corner defined by the
quadrants of two circles), an arc sandwiched between identical quadrants
of two concentric circles?

A continuous straight railroad track of one mile is permanently tied
down at both ends. As the day heats up, the coefficient of expansion
of steel causes the rail to expand so that the length is now 5281
feet. Assuming that the track expands upward, what maximum vertical
distance from the horizontal will the track rise at the highest point?

Show that the point of intersection Q of the axis of the parabola y=x^ 2
and the hypotenuse of right triangle RST (inscribed in the parabola so
that R coincides with the vertex of the parabola) is independent of the
choice of right triangle.

Can two concentric circles share only a few points? If they are
concentric and they have the same radius, they would share all of
their points, and if they don't have the same radius they will share
no points. It seems like it's all or none.

In your FAQ on circle formulas, in the sections where the other five
values are derived from any two known values, could you write each
formula in terms of only the two known values, instead of using the
intermediate steps?